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In mathematics, the Jacobian conjecture is a celebrated problem on polynomials in several variables. It was first posed in 1939 by Ott-Heinrich Keller. It was widely publicized by Shreeram Abhyankar, as an example of a question in the area of algebraic geometry that requires little beyond a knowledge of calculus to state. The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. As of 2015, there are no plausible claims to have proved it. Even the two variable case has resisted all efforts. There are no known compelling reasons for believing it to be true, and according to there are some suspicions that the conjecture is in fact false for large numbers of variables. ==The Jacobian determinant== Let ''N'' > 1 be a fixed integer and consider the polynomials ''f''1, ..., ''f''''N'' in variables ''X''1, ..., ''X''''N'' with coefficients in a field ''k''. Then we define a vector-valued function ''F'': ''kN'' → ''k''''N'' by setting: : ''F''(''c''1, ..., ''c''''N'') = (''f''1(''c''1, ...,''c''''N''),..., ''f''''N''(''c''1,...,''c''''N'')) The Jacobian determinant of ''F'', denoted by ''JF'', is defined as the determinant of the ''N'' × ''N'' Jacobian matrix consisting of the partial derivatives of ''fi'' with respect to ''Xj'': : then ''JF'' is itself a polynomial function of the ''N'' variables ''X''1, ..., ''XN''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Jacobian conjecture」の詳細全文を読む スポンサード リンク
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